{"id":1399,"date":"2025-12-31T07:22:32","date_gmt":"2025-12-31T06:22:32","guid":{"rendered":"http:\/\/mymatma.pl\/?p=1399"},"modified":"2025-12-31T07:23:50","modified_gmt":"2025-12-31T06:23:50","slug":"zadanie-140","status":"publish","type":"post","link":"https:\/\/mymatma.pl\/?p=1399","title":{"rendered":"Zadanie 140"},"content":{"rendered":"\n<p><strong>zadanie 14 \u2013 marzec 2021 (1pkt)<\/strong><\/p>\n\n\n\n<p>Ci\u0105gi (a<sub>n<\/sub>) , (b<sub>n<\/sub>) oraz (c<sub>n<\/sub>) s\u0105 okre\u015blone dla ka\u017cdej liczby naturalnej n\u22651 nast\u0119puj\u0105co:<\/p>\n\n\n\n<p>a<sub>n<\/sub>=6n<sup>2<\/sup>\u2212n<sup>3<\/sup><\/p>\n\n\n\n<p>b<sub>n<\/sub>=2n+13<\/p>\n\n\n\n<p>c<sub>n<\/sub>=2n<\/p>\n\n\n\n<p>Wska\u017c zdanie prawdziwe:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>a) Ci\u0105g (a<sub>n<\/sub>) jest arytmetyczny.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>b) Ci\u0105g (b<sub>n<\/sub>) jest arytmetyczny.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>c) Ci\u0105g (c<sub>n<\/sub>) jest arytmetyczny.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>d) W\u015br\u00f3d ci\u0105g\u00f3w (a<sub>n<\/sub>),(b<sub>n<\/sub>),(c<sub>n<\/sub>) nie ma ci\u0105gu arytmetycznego.<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n    <div class=\"premium-wrapper-odp\" id=\"odp_69e22748e7751\">\n        <button class=\"premium-btn-odp\"\n                data-id=\"odp_69e22748e7751\"\n                style=\"margin-top:10px; padding:12px 20px; background:#28a745; color:#fff; border-radius:25px; border:none; cursor:pointer; font-weight:bold;\">\n            Poka\u017c odpowied\u017a\n        <\/button>\n\n        <div class=\"premium-content-odp\" style=\"display:none; padding:15px; border:1px solid #28a745; margin-top:10px; background:#e6ffe6; border-radius:25px;\">\n            \nPrawid\u0142owa odpowied\u017a to: b) Ci\u0105g (b<sub>n<\/sub>) jest arytmetyczny.\n        <\/div>\n    <\/div>\n    \n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n    <div class=\"premium-wrapper-wyjasnienie\" id=\"wyjasnienie_69e22748e7761\">\n        <button class=\"premium-btn-wyjasnienie\"\n                data-id=\"wyjasnienie_69e22748e7761\"\n                style=\"margin-top:10px; padding:12px 20px; background:#28a745; color:#fff; border-radius:25px; border:none; cursor:pointer; font-weight:bold;\">\n            Poka\u017c wyja\u015bnienie\n        <\/button>\n\n        <div class=\"premium-content-wyjasnienie\" style=\"display:none; padding:15px; border:1px solid #28a745; margin-top:10px; background:#e6ffe6; border-radius:25px;\">\n            \n<h2>Wyja\u015bnienie<\/h2>\n\n<p>Aby sprawdzi\u0107, kt\u00f3ry z podanych ci\u0105g\u00f3w jest arytmetyczny, nale\u017cy zbada\u0107,\nczy r\u00f3\u017cnica mi\u0119dzy kolejnymi wyrazami jest sta\u0142a.<\/p>\n\n<h3>Ci\u0105g <code>(a<sub>n<\/sub>)<\/code><\/h3>\n<p>\na<sub>n<\/sub> = 6n\u00b2 \u2212 n\u00b3 \u2014 jest to wielomian trzeciego stopnia.<br>\nR\u00f3\u017cnice mi\u0119dzy kolejnymi wyrazami nie s\u0105 sta\u0142e, wi\u0119c ci\u0105g <strong>nie jest arytmetyczny<\/strong>.\n<\/p>\n\n<h3>Ci\u0105g <code>(b<sub>n<\/sub>)<\/code><\/h3>\n<p>\nb<sub>n<\/sub> = 2n + 13 \u2014 jest to funkcja liniowa.<br>\nObliczamy r\u00f3\u017cnic\u0119:\n<\/p>\n\n<p>\nb<sub>n+1<\/sub> \u2212 b<sub>n<\/sub> = [2(n+1) + 13] \u2212 (2n + 13) = 2\n<\/p>\n\n<p>\nR\u00f3\u017cnica jest sta\u0142a, wi\u0119c ci\u0105g <strong>jest arytmetyczny<\/strong>.\n<\/p>\n\n<h3>Ci\u0105g <code>(c<sub>n<\/sub>)<\/code><\/h3>\n<p>\nc<sub>n<\/sub> = 2\u207f \u2014 to ci\u0105g geometryczny, poniewa\u017c ka\u017cdy kolejny wyraz\notrzymujemy przez pomno\u017cenie poprzedniego przez 2.<br>\nNie ma sta\u0142ej r\u00f3\u017cnicy, wi\u0119c <strong>nie jest arytmetyczny<\/strong>.\n<\/p>\n\n<h3>Wniosek<\/h3>\n<p>\nJedynym ci\u0105giem arytmetycznym w\u015br\u00f3d podanych jest <strong>(b<sub>n<\/sub>)<\/strong>.\n<\/p>\n\n        <\/div>\n    <\/div>\n    \n<\/div>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>zadanie 14 \u2013 marzec 2021 (1pkt) Ci\u0105gi (an) , (bn) oraz (cn) s\u0105 okre\u015blone dla ka\u017cdej liczby naturalnej n\u22651 nast\u0119puj\u0105co: an=6n2\u2212n3 bn=2n+13 cn=2n Wska\u017c zdanie prawdziwe: a) Ci\u0105g (an) jest arytmetyczny. b) Ci\u0105g (bn) jest arytmetyczny. c) Ci\u0105g (cn) jest arytmetyczny. d) W\u015br\u00f3d ci\u0105g\u00f3w (an),(bn),(cn) nie ma ci\u0105gu arytmetycznego.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"saved_in_kubio":false,"_uag_custom_page_level_css":"","_themeisle_gutenberg_block_has_review":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[25,94,204,81,83],"tags":[23,92,205],"class_list":["post-1399","post","type-post","status-publish","format-standard","hentry","category-ciag-arytmetyczny","category-marzec-2021","category-rozne","category-zadania-maturalne","category-zadania-tematami","tag-ciag-arytmetyczny","tag-marzec-2021","tag-rozne"],"jetpack_featured_media_url":"","uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"kubio-fullhd":false,"woocommerce_thumbnail":false,"woocommerce_single":false,"woocommerce_gallery_thumbnail":false},"uagb_author_info":{"display_name":"adminmymatma","author_link":"https:\/\/mymatma.pl\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"zadanie 14 \u2013 marzec 2021 (1pkt) Ci\u0105gi (an) , (bn) oraz (cn) s\u0105 okre\u015blone dla ka\u017cdej liczby naturalnej n\u22651 nast\u0119puj\u0105co: an=6n2\u2212n3 bn=2n+13 cn=2n Wska\u017c zdanie prawdziwe: a) Ci\u0105g (an) jest arytmetyczny. b) Ci\u0105g (bn) jest arytmetyczny. c) Ci\u0105g (cn) jest arytmetyczny. d) W\u015br\u00f3d ci\u0105g\u00f3w (an),(bn),(cn) nie ma ci\u0105gu arytmetycznego.","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1399","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1399"}],"version-history":[{"count":2,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1399\/revisions"}],"predecessor-version":[{"id":1401,"href":"https:\/\/mymatma.pl\/index.php?rest_route=\/wp\/v2\/posts\/1399\/revisions\/1401"}],"wp:attachment":[{"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1399"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1399"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymatma.pl\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}